When can we exchange order of two limits?

Solution 1:

A simple example of a doubly indexed sequence $a_{m,n}$ for which you cannot exchange limits is given in Rudin's "Principles..." Example 7.2 pg. 144:

Let $a_{m,n} = \frac{m}{m+n}$, then $$\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty}a_{m,n} = 1,$$ but $$\lim_{m\rightarrow\infty}\lim_{n\rightarrow\infty}a_{m,n} = 0.$$

Here is a previous post on this question which seems to thoroughly answer your other questions: When can you switch the order of limits?

Solution 2:

A simpler example where one cannot switch the limits: $$a_{m,n}=\begin{cases}0&\text{if }m\le n\\1&\text{if }m>n\end{cases}$$

Solution 3:

$f(x,y)=\frac{x-y}{x+y}.$

You can also check the limit at the point $(0,0)$. Typical example of calculus.